Exact, {ital E}=0, classical solutions for general power-law potentials

Journal Article · Sun Oct 01 04:00:00 EDT 1995 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PhysRevE.52.4430· OSTI ID:239414

Daboul, J ^[1]; Nieto, M M ^[2]

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1. Physics Department, Ben Gurion University of the Negev, Beer Sheva (Israel)
2. Theoretical Division, Los Alamos National Laboratory, University of California, Los Alamos, New Mexico 87545 (United States)

For zero energy, {ital E}=0, we derive exact, classical solutions for {ital all} power-law potentials, {ital V}({ital r})={minus}{gamma}/{ital r}{sup {nu}}, with {gamma}{gt}0 and {minus}{infinity}{lt}{nu}{lt}{infinity}. When the angular momentum is nonzero, these solutions lead to the orbits {rho}({ital t})=(cos{l_brace}{mu}[{ital cphi}({ital t}){minus}{ital cphi}{sub 0}({ital t})]{r_brace}){sup 1/{mu}}, for all {mu}{equivalent_to}{nu}/2{minus}1{ne}0. When {nu}{gt}2, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of {ital cphi}({ital t}) and {ital cphi}{sub 0}({ital t}), as functions of {ital t}, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. In addition to the special {nu}=2 case, the unbound trajectories are also discussd in detail. This includes the unusual trajectories which have finite travel times to infinity.

• OSTI ID: 239414
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 4 Vol. 52; ISSN PLEEE8; ISSN 1063-651X
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
ANALYTICAL SOLUTION
ANGULAR MOMENTUM
BOUNDARY CONDITIONS
CENTRAL POTENTIAL
CLASSICAL MECHANICS
QUANTUM MECHANICS
TRAJECTORIES